Tony Hawk's Proving Ground is made up of specific goals for each of these classes. While completists will max out all three, the class you're favoring at key moments shapes you, your eventual team, and other important factors. The best part, though, is how most of the new mechanics in the game tailor to a specific class.
We all loved the slow-mo, detailed Nail the Trick from Project 8. It's only natural, then, that Proving Ground would let you Nail the Grab and Nail the Manual on top of it. While this may result in you spending more play time in slow-motion than regular speed, the precision and creativity that go into tricks and combos as a result are staggering.
Next are those class-geared goodies. For Hardcore skaters, Agro Kick, Skate Checking, and Bowl Carving allow you to go bigger, ollie higher, and let nobody stand in your way. Riggers have a bevy of skills to modify the world, basically giving you create-a-park privileges in the existing world. Drop a railing anywhere you please, or even modify existing objects to suit your twisted desires.